Tokyo Polytechnic University Repository / 東京工芸大学学術リポジトリ

18KJ00004761222作品Workdepartmental bulletin pape

甲賀, 正彦

2789

コウガ, マサヒコ

KOGA, Masahiko

2790

Poster, Three Dimensional Illustration

8KJ00004878586論文Articledepartmental bulletin pape

小沢, 一仁

1779

オザワ, カズヒト

滝沢, 利直

1780

タキザワ, トシナオ

菅田, 圭次

1781

スガタ, ケイジ

Ozawa, Kazuhito

1782

Takizawa, Toshinao

1783

Sugata, Keiji

1784

A Study of the Relationship between Students' Self-understanding and their Social Recognition(5): Necessity and Possibility supporting Engineering Education of Formation of "the Power of Living"

1KJ00004878589論文Articledepartmental bulletin pape

前原, 和寿

マエハラ, カズヒサ

Maehara, Kazuhisa

1165

Non Commutative Geometry 1
Kazuhisa MAEHARA∗
Abstract
In this article we shall introduce a non commutative algebraic geometry by Kont-
sevich and Rosenberg([Kon], [Mch]) and represent it by recently develoved theory of
corings and comodules([Brz]). We restrict ourselves to the category of non commu-
tative algebraic varieties and develope the birational geometry by inﬁnite Galois the-
ory of skew ﬁelds making use of proﬁnite groups([AM],[BJ],[Breen1],[Breen2],[Gir],
[SGA], [S1], [S2], [Shatz], [Se], [Zuo], [RBZL]). We apply it to non commutative
varieties of general type deﬁned later over the ﬁeld of characteristic 0([Iita], [Fuj],
[Kaw], [Mats], [MP], [Km3]). Main tools of classiﬁcation of projective varieties([Iita],
[Mum], [Vieh], [Ko1], [Zuo]) are so called characteristic p > 0 technic([MP], [Ko2]),
[BBD], [Berth]) and weak positivity direct images of multi-power of dualizing sheaves
for ﬁbre spaces([Kaw], [Ws], [Vieh], [Nak], [Km1]) as well as Kawamata-Viehweg
vanishing theorems([MP]). Instead of these tools, we make use of proﬁnite groups.
1 Introduction:
In this section we consider the corings([Brz])that have a grouplike element g which are
related to ring extensions B → A. Throughout this section C denotes an A-coring. Galois
corings are isomorphic to the Sweedler coring associated to a ring extension B → A
induced by the existence of a grouplike element. The following theorem determines when
the g-coinvariants functor is an equivalence.
Theorem 2. Let g be a grouplike element of C, B = AcoCg , and Gg : M
C → MB M →
M coCg the g-coinvariants functor.
1. The following statements are equivalent:
(i) (C, g) is a Galois coring and A is a ﬂat left B-module.
∗Associate Professor, General Education and Research Center, Tokyo Polytechnic University
Received Sept.18, 2007
1 
:maehara gen.t-kougei.ac.jp.,E-mail:
ACADEMIC REPORTS Fac. Eng. Tokyo Polytech. Univ. Vol. 30 No.1 (2007) 
(ii) AC is ﬂat and Ag is a generator in M
C.
2. The following statements are equivalent, too.
(i) (C, g) is a Galois coring and BA is faithfully ﬂat.
(ii) AC is ﬂat and Ag is a projective generator in M
C.
(iii) AC is ﬂat and Hom
C(Ag,−) : MC → MB is an equivalence whose inverse is
−⊗B A : MB →MC.
The theorem above is a restatement of one of the main results in non commutative descent
theory([HS], [S1], [S2], [Km2]). In fact, for an algebra extension B → A, there exists a
comparison functor − ⊗B A : MB → Desc(A/B) which to each right B-module M gives
a descent datum (M ⊗B A, f) with f : M ⊗B A→M ⊗B A⊗B A,m⊗ a → m⊗ 1A ⊗ a.
If (C, g) is a Galois coring, then the category of right C-comodules is isomorphic to the
category of descent data Desc(A/B). Thus if B → A is faithfully ﬂat, then it is an eﬀective
descent morphism. Furthermore, Galois corings correspond to comparison functors that
are equivalences. Note that if B → A is a faithful ﬂat extension, then (A⊗B A, 1A⊗B 1A)
is a Galois coring. The objects in the category of corings are pairs (C : A), where A is an
R-algebra and C is an A-coring. A morphism between corings (C : A) and (D : B) is a
pair of mappings (γ : α) : (C : A)→ (D : B) satisfying
1. α : A→ B is an algebra map. Hence D is considered to be an (A,A)-bimodule.
2. γ : C → D is a map of (A,A)-bimodules such that
ξ ◦ (γ ⊗A γ) ◦ΔC = ΔC ◦ γ, εD ◦ γ = α ◦ εC ,
where ξ : D ⊗A D → D ⊗B D is the canonical map of (A,A)-bimodules.
Since an algebra A can be considered as a trivial A-coring (A : A), this category of corings
contains the category of R-algebras.
Left C-comodule is deﬁned as a left A-module M , with a coassociative and counital
left C-coaction.C-morphisms between left C-comodules M , N are deﬁned in an obvious
way. Left C-comodules and their morphisms form a pre-additive category CM .
3 Geometric View
Let k be a commutative ﬁeld and A, B k-algebras. The objects of the opposite category
of corings denote Spec (C : A) and a morphism between Spec (D : B) → Spec (C : A)
denotes Spec (γ : α). This category is said to be that of covers. Furthermore, the
2 ACADEMIC REPORTS Fac. Eng. Tokyo Polytech. Univ. Vol. 30 No.1 (2007) 
category CM is abelian and it is denoted QCoh(Spec (C : A)). The canonical morphism
f : Spec (B ⊗A B : B) → Spec (A : A) deﬁnes an equivalence between abelian categories
f∗ : QCoh(Spec (A : A)) ∼= QCoh(Spec (B ⊗A B : B)). Owing to Morita-Takeuchi
theorems or Grothendieck ideas, the geometry of covers consist in QCoh(Spec (C : A)).
The cover Spec (C : A) equipped with an epimorphism A⊗A→ C which is a morphism
of coalgebras is said to be a space cover. A morphism in the category of space covers
is deﬁned to be a morphism as covers compatible with additional structure as space
covers. Let f = (γ, α), g = (δ, β) be two morphisms between space covers Spec (C :
A) → Spec (D : B). When for xi ⊗ yi ∈ ker(A ⊗ A → C, the following equation holds
∑
i α(xi)·β(yi)−β(xi)·α(yi) = 0 in B, two morphisms f and g are deﬁned to be equivalent.
Deﬁnition 4. The category of non commutative algebraic spaces over k is the localization
category with the canonical morphisms invertible of the quotient of the category of space
covers by equivalence of equivalent morphisms.
The category of separated quasi-compact schemes over k([Gir], [SGA], [GG], [HS],
[Kato], [KKMS]) and the opposite category of that of k-algebras are equivalent to a full
subcategory of the category of non commutative algebraic spaces over k([Kon], [Cohn]),
respectively. The category of non commutative algebraic spaces over k admits ﬁnite limits.
A non commutative algebraic space of the type Spec (A : A), where A is a k-algebra, is
said to be an aﬃne space. Let NPd−1k be the non commutative projective space over k
and A a k-algebra. The set Hom(Spec (A : A), NPd−1k ) is the set of quotient modules of
Ad which are locally free A-modules of dimension 1 in ﬂat topology([SGA]). In the same
way, we have the non commutative Grassmannian NGrk(r, d)([Kon], [Laum]).
5 Extension of skew ﬁelds and Galois theory
Let A be an integral domain such that xA ∩ yA 
= 0 for x, y ∈ A, which is called a right
Ore domain. Let S = R×. Then the localization of A at S is a skew ﬁeld K = AS and
the natural homomorphism λ : A → K is a monomorphism. Racall that every ring with
a homomorphism to a ﬁeld has invariant basis number. From now on, we treat a non
commutative algebraic space of the type Spec (C : A) where A is a Ore domain. Any
equation of degree n > 0, xn + a1x
n−1 + · · · + an = 0 (ai ∈ K), has a right root in some
extension of K. There exists the right algebraic closure K over K such that any equation
of the type above has a right root in K. A Galois extension L/K is outer if and only
if the centralizer of K in L is just the centralizer of L. Let k be a commutative ﬁeld of
characteristic 0 and K a k-algebra of ﬁnite type, skew ﬁeld. Let K be the right algebraic
3 Non Commutative Geometry 1 
closure of K such that the centralizer of K in K is just the centralizer of K([Cohn]). Let
(Ki)i∈I be a family of skew ﬁelds such that
1. Ki are subﬁelds of K,
2. Ki are k-algebras of ﬁnite type,
3. the centralizers of Ki in K are the center of K.
Then the K/Ki are all outer Galois extensions, whose Galois groups are proﬁnite groups.
We need Jacobson-Bourbaki correspondence([Cohn], [BJ]): Let K be a ﬁeld and End(K)
the endomorphism ring of the additive group K+ with the ﬁnite topology. We have an
order-reversing bijection between the subﬁelds D of K and the closed K-subrings of the
type EndD−(K) of End(K). From this, we have the following Galois connection: Let L/K
be an algebraic Galois extension with Galois group G outer. Then we have a bijection
between intermediate ﬁelds D, i.e., K ⊂ D ⊂ L and the closed subgroups H.
6 Non commutative algebraic birational geometry
We investigate the non commutative algebraic birational geometry from the point of
view of the proﬁnite Galois groups([Gir], [Breen1], [Breen2]). Let X → S be a non
commutative ﬁbre space of algebraic spaces over Spec (k), with the generic point of the
generic general ﬁbre one of skew ﬁelds Ki which are deﬁned in the preceding section([Mch],
[RBZL]). Let 1 → G → E → P → 1 be an extension of a proﬁnite group P by a
proﬁnite group G associated to the non commutative ﬁbre space X → S. Hence G is
a proﬁnite group, that is one of the Galois group Gal(K/Ki). To an exact sequence
1→ InnG→ AutG→ OutG→ 1, we have an exact sequence
H1(P, InnG)→ H1(P,AutG)→ H1(P,OutG),
i.e.,
Hom(P, InnG)→ Hom(P,AutG)→ Hom(P,OutG).
Here OutG denotes the outer automorphism group of G. A group extension is an element
of H1(P,G→ AutG) , where G→ AutG is a crossed module. We have
1→ H2(P,Z(G))→ H1(P,G→ AutG)→ H1(P,OutG).
Here Z(G) denotes the center of G. Assume that Out(G) is an algebraic group of countable
connected components. Then the canonical representation ρ : P → OutG turns out to
be trivial after replacing a proﬁnite group associated to a ﬁnite morphism S ′ → S in the
4 ACADEMIC REPORTS Fac. Eng. Tokyo Polytech. Univ. Vol. 30 No.1 (2007) 
following lemma. Furthermore assume that the extension is neutral. This assumption
is satisﬁed since there exists a homomorphism from 1 → G′ → G′ × P → P → 1 to
1→ G→ E → P → 1, where P → P is an identity, G′ = Gal(K/K).
Since we have H2(P,Z(G)) → H1(P,G → Aut(G)), the extension 1 → G → E →
P → 1 is given by pushing out an extension 1 → Z(G) → E′ → P → 1. Hence E ′
is a semi-direct product Z(G)  P , which is contained in a semi-direct product G  P .
Thus this central extension is trivial. Therefore by pushing out this central extension, the
extension 1→ G→ E → P → 1 is trivial.
Lemma 7. There exists a homomorphism P ′ → P with (P ′ : P ) < ∞ such that the
representation ρ : P ′ → Out(G) is trivial. Here P ′ denotes the absolute Galois group
Gal(R(S ′)/R(S ′)).
Proof. Let A denote Out(G). This group A is locally algebraic([SGA]). The natural
representation ρ : P → A induces ρ : P → A/A0, where A0 denotes the neutral component
of A. There is no countable proﬁnite group. Since A/A0 is a countable set, ρ(P ) is a
ﬁnite group. Replace by P the kernel of ρ. We have ρ : P → A0. Hence we have an
isomorphism
H1(R(S)/R(S), A0(R(S)) ∼= H1(BP,A0).
Let P be an A0-torsor associated to ρ : P → A0. A0 is algebraic (quasi-compact, faithfully
ﬂat and of ﬁnite type) over Spec (R(S)). Thus there exists a generically ﬁnite S ′ → S such
that an A0-torsor P is trivial over Spec (R(S ′)). Hence the representation ρ : P ′ → Out(G)
is trivial.
Thus we obtain the following result in our proof.
Theorem 8. Let 1 → G → E → P → 1 be an extension of a proﬁnite group P by a
proﬁnite group G. Assume
(a) Out(G), is an algebraic group with countable connected components.
(b) E → P has a section which is a group homomorphism, i.e., a neutral extension.
Then there exists a proﬁnite group P ′ such that the pull-back of the extension 1→ G×P
P ′ → E ×P P ′ → P ′ → 1 is a direct product.
Let X be a non commutative ﬁbre space of smooth varieties over Spec k. We have
the canonical homomorphism Γ(X,Ω⊗mX ) ⊗ OX → Ω⊗mX . Assume this homomorphism is
generically epimorphism. Then it determines a map from an open of X to non commu-
tative Grassmannian[Kon]). When this map is birational, i.e., the ﬁeld deﬁned by the
generic point of X and that of the image are isomorphic, the assumption (a) above is
satisﬁed.
5 Non Commutative Geometry 1 
Remark 9. Let φ : G1 → G2 be an open continuous homomorphism of proﬁnite groups.
φ(G1) ⊂ G2. Let Z(G2)Cφ(G1)(φ(G1)) denote C. Then for a homomorphism between
extensions of P by G1 and G2 respectively, one has homomorphisms H
2(P,Z(G1)) →
H2(P, φ(Z(G1))) → H2(P,C) . There exists an open subgroup P ′ of ﬁnite index of P
such that H2(P ′, Z(G2))→ H2(P ′, C) is injective.
References
[AM] Adem, A. and Milgram, R., Cohomology of ﬁnite groups., Grundlehren math. Wiss.
309, p. 317 (1991).
[BBD] Beilinson A.A., Bernstein I.N., Deligne P., Faisceaux pervers., Analyse et Topolo-
gie sur les espaces singuliers(I), Confe´rence de Luminy, juillet 1981, Aste´risque
100(1982).
[Berth] Berthelot, P., Alte´rations de varie´te´s alge´braiques d’apre`s A.J. DE JONG,
Se´minaire Bourbaki 48, n◦ 815, pp.273-311(1997).
[BJ] Borceux, F. and Janelidze, G. Galois Theories., cambridge studies in advanced
math. 72 pp.341 ISBN 0 521 80309 8 2001
[Breen1] Breen, L., The´orie de Schreier supe´riure., Ann. scient. E´c. Norm. Sup., 4e se´rie,
t. 25, pp. 465-514(1992).
[Breen2] Breen, L., On the classiﬁcation of 2-gerbes and 2-stacks, 225 Aste´risque Socie´te´
Mathe´matique p. 160 (1994).
[Brz] Brzezinski, T., Wisbauer, R., Corings and Comodules, London Mathematical Soci-
ety Lecture Note Series 309, Cambridge University Press, 2003.
[Cohn] Cohn, P.M., Skew Fields, Theory of general division rings, Encyclopedia of
mathematics and applications 57.
[Fuj] Fujita, T., On Ka¨hler ﬁbre spaces over curves., J. Math.Soc. Japan 30, pp. 779-
794(1978).
[Gir] Giraud, J., Cohomologie non abe´lienne., Grundlehren math. Wiss., Springer-Verlag
Berlin Heidelberg New York, p. 467,(1971).
[SGA] Grothendieck, A. et al., Se´miaire de ge´ome´trie alge´brique du Bois-Marie, SGA1,
SGA4 I,II,III, SGA41/2, SGA5, SGA7 I,II, Lecture Notes in Mat., vols. 224,269-270-
305,569,589,288-340, Springer-Verlag, New York, 1971-1977.
6 ACADEMIC REPORTS Fac. Eng. Tokyo Polytech. Univ. Vol. 30 No.1 (2007) 
[GG] Grothendieck, A., Fondaments de la ge´ome´trie alge´brique., Secre´trariat
mathe´matique, 11 rue Pierre Curis, Paris 5e, p. 236 (1962).
[HS] Hirshowits, A. and Simpson, C., Descente pour les n-champs., arXiv:math.
AG/9807049v3 13 Mar 2001
[Iita] Iitaka, S., Introduction to birational geometry., Graduate Textbook in Mathematics,
Springer-Verlag, p. 357 (1976).
[Laum] Laumon, G., Moret-Bailly, L., Champs alge´braiques., Universite´ de Paris-sud
Mathe´matiques, p. 94 (1992).
[Kato] Kato, K., Logarithmic structures of Fontaine-Illusie., Algebraic Analysis, Ge-
ometry and Number Theory., The Johns-Hopkins Univ. Press, Baltimore, MD, pp.
191-224(1989).
[Kaw] Kawamata, Y., Minimal models and the Kodaira dimension of algebraic ﬁbre
spaces., J. Reine Angew. Math. 363, pp. 1-46 (1985).
[KKMS] Kemf, G., Knudsen, F., Mumford, D., Saint-Donat, B., Troidal embeddings I.,
Springer Lecture Notes, 339(1973).
[Ko1] Kolla´r, J., Shafarevich maps and automorphic forms., Princeton university press,
Princeton, New Jersey pp.201 1995.
[Ko2] Kolla´r, J., Rational curves on algebraic varieties., Springer, Berlin-Heiderberg-
Newyork-Tokyo, (1995)
[Kon] Kontsevich, M., Rosenberg A. Noncommutative smooth spaces.,
arXiv:math.AG/9812158v1 30Dec 1998.
[Km1] Maehara, K., Diophantine problems of algebraic varieties and Hodge theory in
International Symposium Holomorphic Mappings, Diophantine Geometry and related
Topics in Honor of Professor Shoshichi Kobayashi on his 60th birthday., R.I.M.S.,
Kyoto University October 26-30, Organizer: Junjiro Noguchi(T.I.T.), pp. 167-187
(1992).
[Km2] Maehara, K., Algebraic champs and Kummer coverings., A.C.Rep. T.I.P. Vol.18,
No.1, pp.1-9(1995).
[Km3] Maehara, K., Conjectures on birational geometry., A.C.Rep. T.I.P. Vol.24, No.1,
pp.1-10(2001).
7 Non Commutative Geometry 1 
[Mats] Matsuki K., Introduction to the Mori Program., Universitext p. 468 Springer 2000
[MP] Miyaoka Y., Peternel T., Geometry of Higher Dimensional Algebraic Varieties.,
DMV Seminar Band 26 Birkha¨iser p. 213 1997
[Mch] Mochizuki S., The local Pro-p Anabelian Geometry of Curves., Research Institute
for Mathematical Sciences, Kyoto University RIMS-1097(1996).
[Mum] Mumford D., Selected Papers: On the classiﬁcation of varieties and moduli
spaces., Springer p. 795 2003
[Nak] Nakayama N., Invariance of the plurigenera of algebraic varieties under minimal
model conjectures, Topology 25(1986) 237-251.
[RBZL] Ribes L., Zalesskii P., Proﬁnite Groups., Ergebnisse der Mathematik und ihrer
Grenzgebiete 3.Folge Vol.40 Springer(1991).
[Ws] Schmid, W., Variation of Hodge structure: the singularities of the period mapping.,
Inventiones math., 22., pp. 211-319(1973).
[Se] J-P. Serre. Cohomologie Galoisienne. Lecture Notes in Mathematics. , 5 4th Ed. 1973.
[S1] Simpson, C., Algebraic (geometric) n-stacks., arXiv:math. AG/9609014
[S2] Simpson, C., A closed model structure for n-categories, internal Hom, n-stacks and
generalized Seifer-Van Kampen., arXiv:math: AG/9704006 v2 17 Mar 2000
[Shatz] Shatz, S., Proﬁnite groups, arithmetic, and geometry., Annals of Math. studies,
N. 67, Princeton Uni. Press, p. 252,(1972).
[Vieh] Viehweg, E. Quasi-projective Moduli for Polarized Manifolds., Ergebnisse der
Mathematik und ihrer Grenzgebiete, 3.Folge.Band 30, p. 320 (1991).
[Zuo] Zuo, K., Representations of fundamental groups of algebraic varieties., Lecture
Notes in Math. 1708, Springer, ISBN 3-540-66312-6.
8 ACADEMIC REPORTS Fac. Eng. Tokyo Polytech. Univ. Vol. 30 No.1 (2007) 


Non Commutative Geometry I

5KJ00004878583論文Articledepartmental bulletin pape

吉田, 昌弘

1769

ヨシダ, マサヒロ

Yoshida, Masahiro

1770

Modern Rationalism and The Medical Crime of Nazi Germany

12KJ00004761216論文Paperdepartmental bulletin pape

今村, みゑ子

2779

イマムラ, ミエコ

IMAMURA, Mieko

2780

Tobasojo Kakuyu and Choju Giga

編集後記

5KJ00004878594論文Articledepartmental bulletin pape

藤橋, 忠悟

3216

フジハシ, チュウゴ

行谷, 時男

ユキヤ, トキオ

Asenov, Asen

3287

Fujihashi, Chugo

1175

Yukiya, Tokio

1176

1177

Electron and Hole Current Switching n-i-p-Type Semiconductor Quantum Dot Transistor

9KJ00004878587論文Articledepartmental bulletin pape

八木橋, 宏勇

1785

ヤギハシ, ヒロトシ

Yagihashi, Hirotoshi

1786

A Cognitive Approach to Idiom Learning II

表紙のことば

10KJ00004761214論文Paperdepartmental bulletin pape

白倉, 克文

2775

シラクラ, カツフミ

SHIRAKURA, Katsufumi

2776

Comparative Thoughts on the Histories of Music in Japan and Russia

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